Hausdorff dimension for exactly approximable matrices

Source abstract

We establish the Hausdorff dimension for exactly approximable matrices for all dimensions, including the inhomogeneous case. More precisely, let $ψ:\N\to(0,\infty)$ be a non-increasing approximation function and let $τ_ψ$ denote its lower order. For $m,n\ge1$ and $\boldsymbolθ\in\mathbb R^m$, denote $\Exact_{m,n}(ψ,\boldsymbolθ)$ the set of $m\times n$ matrices that are $ψ$-approximable with shift $\boldsymbolθ$, but are not $cψ$-approximable for any $0 \frac{n}{m}. \] We adopt a probabilistic viewpoint, i.e. by an averaged counting argument to quantify the distribution of rational vectors and subspaces $$\left\{\frac{\mathbf p+\boldsymbolθ}{q}: (\mathbf p, q)\in \mathbb{Z}^{m+1}\right\},\ \ \ \bigg\{\big\{A\in \Mat_{m, n}(\R): A\mathbf q-\mathbf p-\boldsymbolθ=0\big\}: (\mathbf p, \mathbf q)\in \mathbb{Z}^{m+n}\bigg\}.$$